algorithmic design and analysis of architectural origami
TRANSCRIPT
RESEARCH
Algorithmic Design and Analysis of ArchitecturalOrigami
Mia Tsiamis1 • Alfonso Oliva1 • Michele Calvano2
Published online: 21 December 2017
� Kim Williams Books, Turin 2017
Abstract The objective of this research is to explore the ability of folded origami
tessellations to adapt to free-form surface configurations and to evaluate the
structural behavior of these folded tessellations for architectural and engineering
applications. A mathematical exploration of the kinematics of folding provides the
basis for understanding the relationship between the geometric parameters of the
Miura-Ori origami fold pattern and the flexibility of the three-dimensional folded
form. The structural behavior of folded plate structures derived from the Miura
pattern on an architectural scale is studied in an effort to answer the question: does
function follow fold? A parametric workflow provides the platform for evaluating
the effect of fold pattern variations on the structural performance of the folded
geometry, and computational methods are used to optimize the performance of the
structure.
Introduction
The geometric flexibility of folded origami tessellations, combined with the
structural efficiency of folded plate structures, gives origami strong potential for
architectural and engineering applications. The flexibility of the folded Miura
pattern can be attributed to the auxetic nature of the fold pattern. This auxetic nature
implies that the folded form has a negative Poisson ratio, so that when the folded
tessellation is pulled along one axis, it will expand along the orthogonal axis. This
& Mia Tsiamis
1 Leslie E. Robertson Associates (LERA?), 40 Wall Street, 23rd Floor, New York, NY 10005,
USA
2 Department of History, Design and Restoration of Architecture, Sapienza University, Piazza
Borghese 9, 00186 Rome, Italy
Nexus Netw J (2018) 20:59–73
https://doi.org/10.1007/s00004-017-0361-9
enables the folded tessellation to be globally distorted into shapes with both single
and double curvature, unlike the flat sheet that the tessellation was folded from.
Compared to the unfolded flat sheet, the folded Miura tessellation not only
demonstrates increased deformability, but also increased structural stiffness, as a
result of the increased second moment of area due to the folds in the sheet.
Researchers, including Tomohiro Tachi and Mark Schenk, have explored this
dichotomy of the geometric flexibility and structural stiffness of folded origami
tessellations in their studies of the mechanical properties of origami and the
kinematics of folding. In his work, Tachi has established mathematical rules in order
to simulate folding and to design free-form shapes from origami (Tachi 2009), while
Schenk has examined the process of folding in order to better understand how the
increased stiffness of folded sheets can be used for engineering applications. The
research summarized in this paper aims to build upon the work of Tachi and Schenk
by exploring how folded plate structures derived from the Miura tessellation can be
used to design free-form architectural surface structures.
The research examines the relationship between the geometric parameters of the
Miura fold pattern and the structural performance of the associated folded plate
structure in order to establish the strengths and weaknesses of the pattern and the
viability of using the folded tessellation as a structural system to support free-form
surface geometries on an architectural scale. The research was a collaborative effort
between architects at La Sapienza University in Italy and computational designers
and structural engineers at Leslie E. Robertson Associates in New York, and was
performed in two phases. The first part of the research explored the geometric and
mathematical rules of folding. Visual scripting using Grasshopper, a parametric
modelling plug-in for the 3D modelling software Rhino, was used to generate
origami-based architectural folded plate structures. Parametric modelling techniques
were used to deform the digital representations of the folded Miura tessellation to
match the curvature of various mathematical surfaces, including the doubly curved
paraboloid and hyperbolic paraboloid.
In the second part of the research, the structural behavior of these origami-based
folded plate structures was evaluated in order to gain an understanding of how
variations in the fold pattern, as well as global shape parameters and structural
support conditions, affect the strength and stability of the overall folded form. An
integrated workflow was established to digitally generate the 3D geometric
representation of an origami-inspired folded plate structure, to analyze the structure
under anticipated loading conditions, and to optimize the performance of the
structure by adjusting the distribution of material over the plate structure.
Research Part 1: Geometry and Kinematics of the Origami Unit Module
Origami tessellations are built up from a unit module which is arrayed across a
surface. The geometry of this unit module and the kinematics of folding the module
dictate the range of distortions that can be achieved by the folded origami
tessellation. The unit module is composed of rigid panels that are connected along
their edges by hinges, which enable movement of the panels relative to one another.
60 M. Tsiamis et al.
With an understanding of the kinematics of the unit module, the movement of one
module can be digitally simulated and, subsequently, digital models of the entire
surface pattern can be controlled. Two methods for controlling the folding
simulation of the unit module, and consequently the deformation of the surface
pattern, were developed: (1) The first method uses an algorithm based on geometric
rules to generate a folded form from a flat crease pattern, and (2) the second method
employs a physics engine to simulate folding.
Folding by Geometric Rules
A folded origami tessellation is created by repeating the basic folded unit module in
an array over a surface to create a network of mountain and valley edges which act
as hinges to allow the model to move and achieve various geometric configurations
(Fig. 1). Mountain edges (shown in green in the accompanying figures) are assumed
to move in a positive direction in space, while valley edges (shown in red) are
assumed to move in a negative direction in space relative to the start position. The
relative movement of these edges as the flat pattern is folded is based on the
geometric relationship between the mountain and valley lines of the flat fold pattern.
The fold pattern shown in Fig. 1 represents the unit module of the Miura
tessellation. The designation of mountain and valley lines will result in a standard
‘reverse fold.’ The following geometric relationships and movement rules were used
to develop an algorithm that simulates the folding of a digital model of the reverse
fold:
• Panel S1 will move in direction V1 and panel S2 will synchronously move in
direction V2
• Edges EF and HG move closer together, causing edge IL to move upward (since
it is designated as a mountain fold line) as panels S3 and S4 rotate about edges
EF and HG. Edges EF, HG, and IL remain parallel to one another during this
transformation.
Fig. 1 The module surface; geometric transformation based on rotation about fold lines
Algorithmic Design and Analysis of Architectural Origami 61
• Point P is shared by panel S3 and S4, and K1 and K2 are the projections of P
onto edges EF and HG, respectively. Point P moves simultaneously along
circular paths c and d, which are centered on points K1 and K2, respectively.
There are two solutions to the intersection of Paths c and d. The solution that is
selected to describe the location of point P is based on the designation of hinge
line IL as either mountain or valley fold line.
A digital model was built in the 3D modelling software Rhino and the
Grasshopper plug-in for Rhino was used to implement the algorithm and simulate
folding. Folding of the model is initiated by imposing a translation of point E along
vector V1. This translation then drives changes in the orientation of panels S1, S2,
S3, S4, and S5. Figure 1b illustrates the movement trajectories of the panels. Panel
S1 moves as a result of translation of point E along vector V1. The movement of
panel S3 is the result of translation of point E along vector V1 and rotation about
hinge EF. The movement of panel S5 is the result of translation along vector V1,
rotation about hinge EF, and rotation about hinge EI.
Folding by Physics Simulation
In the second approach to folding simulation, the digital model of the unit module in
Rhino is converted into a network of spring elements using Kangaroo, a physics
engine that runs within the Grasshopper environment. Kangaroo is also used to
apply virtual forces, as well as translational and rotational restraints to the spring
model in order to simulate folding and control the overall configuration of the
folded pattern. With the use of visual programming in Grasshopper, an algorithm is
developed to parametrically generate the flat fold pattern of the tessellation and
subsequently impose bending forces on the surface model in order to generate a 3D
folded shape. The subsequent sections discuss in more detail how the simulation
process was used to fold and deform digital models of the folded Miura tessellation.
Digital Folding of the Miura Unit Module
The first step is to build the unit module in the 3D modelling software Rhino, using
Grasshopper (Fig. 2a). The unit module consists of four quadrilateral panels,
defined by mountain and valley edges. The designation of the fold lines as
mountains and valleys is based on the reverse fold mechanism, in which folding
occurs simultaneously about two orthogonal axes (Fig. 2b).
The unit module was modelled in a partially folded state in order to facilitate
application of the algorithm, which applies forces to generate movement. This unit
module is then repeated over a surface area to build the tessellation. Any duplicate
edges in the resulting array are eliminated to ensure that each panel in the surface
pattern shares a single edge with adjacent panels. The mountain and valley edges of
the partially folded tessellation are polylines v and w (heretofore referred to as ‘zig-
zags’ and ‘strips’, respectively) (Fig. 2b). In order to simulate movement of the
surface pattern in digital space, the edges of the geometric model must be converted
to spring elements, with an assigned stiffness value. This is achieved using
62 M. Tsiamis et al.
Kangaroo, a physics engine that operates within Grasshopper. The entire folded
surface can then be activated by applying a bending force to the hinges between
adjacent faces of the folded mesh. Varying the fold angles generates movement
throughout the entire model. Controlling the degree of this angle is one method that
can be used to control the shape of the folded surface. By applying ‘‘attraction’’
forces between the lowest or highest vertices of the folded surface pattern and a
target surface, these vertices of the fold pattern will be pulled and guided to match
the curvature of the target surface (Fig. 3).
In order to fold the origami pattern, Kangaroo must balance the effect of several
forces acting on the system of springs defined by the fold pattern, including the
stiffness of the springs, bending force about hinges, planarization of mesh faces, and
the ‘PowerLaw’ force that pulls the folded origami to the target surface. During the
development of the folding simulation script, the strengths with which each of these
different forces was applied to the mesh model were varied in order to find the
combination that yielded a correctly folded mesh. Based on the results of testing
wmountain fold
valley fold
neutral fold
(a) (b)
Fig. 2 a Surface pattern with zig-zag and strip polylines highlighted; b Miura unit module
Fig. 3 Miura surface pattern pulled to target surfaces
Algorithmic Design and Analysis of Architectural Origami 63
combinations of force strength values, it can be concluded that there must be one
governing force. For example, if the strength given to the mesh stiffness is high, the
strength given to the PLaw force should be relatively low. It should also be noted
that this difficulty in balancing the effect of forces on the spring system occurs
because the quad-based Miura pattern is over constrained. When the quadrilateral
faces of the pattern mesh are triangulated, the system is no longer over-constrained,
and the mesh folds correctly, regardless of the strengths of the applied forces.
If the tessellation is assumed to be rigidly foldable, all of the quadrilateral panels
would be able to fold continuously from a flat to a fully folded state without
deformation of the panels, and only hinging about the fold lines (Schenk 2011). A
rigidly foldable tessellation will exhibit only one degree of freedom in its movement
range when forces are applied to distort the tessellation. However, if the
quadrangular mesh faces are triangulated by introducing an additional neutral
fold-line along the shorter diagonal of each quadrilateral face (Casale and Valenti
2013), small deformations in the panels will enable the tessellation to adapt its
overall shape to a doubly curved surface under the influence of applied twisting or
bending forces. In order to accurately simulate the physical behavior of the folded
Miura surface pattern under distortion forces, additional forces must be applied to
the system to prevent stretching of the mesh faces and to control planarity of the
panels. Mesh stretching is controlled by limiting all edges of the triangulated mesh
to a constant length. Planarity of the mesh faces is achieved by controlling the
degree of bending about the neutral fold line on each mesh face. If rigid foldability
is to be preserved, bending about this neutral fold should not occur, and the fold
angle should be 180�. When the folded surface pattern is adapted to a continuous,
doubly curved target surface, the bend angle about the neutral fold line can be
reduced to enable further flexibility of the surface pattern.
It was observed that the folding simulation of the quad-based mesh worked much
more smoothly when the target surface was a hyperbolic paraboloid or an arch (bent
about the longitudinal—‘‘strip’’—axis). This may be because the natural physical
response of the Miura pattern is to assume a hyperbolic paraboloid shape when
twisted and a saddle shape when bent (Fig. 4). It is expected that pulling the Miura
pattern to an arch target shape (as well as a paraboloid) will cause some distortion in
the panels because physical models show that the folded Miura is more easily
twisted into a saddle shape (negative Gaussian curvature) than it is bent into an arch
shape (zero Gaussian curvature).
Research Part 2: Structural Evaluation of Architectural Origami
Once the method of generating folded tessellations that match target surfaces was
established, these digital representations of origami-based surface structures were
analyzed as architectural structures to understand their inherent strengths and
weaknesses. Pattern densities of mountain and valley fold-lines, as well as global
shape parameters, such as Gaussian curvature, were varied and the structural
behavior of the folded tessellation was assessed.
64 M. Tsiamis et al.
The structural analysis aimed to determine the effect of both global geometric
parameters, including span, curvature, and support condition, as well as origami
pattern parameters, including ‘zig-zag’ and ‘strip’ density, on the structural behavior
of the folded origami when pulled to match a target surface. Three target surfaces
were studied: paraboloid (positive double curvature), hyperbolic paraboloid
(negative double curvature), and parabolic arch (single curvature). The support
conditions were also varied in order to study variations in the force flow through the
origami structure and to potentially identify a strong and weak direction of the
folded origami structure. Table 1 summarizes the models that were studied.
A parametric workflow provided the platform for evaluating the effect of fold
pattern variations on the structural performance of the folded geometry, and
computational methods were used to optimize the performance of the structure by
varying pattern geometry. The following software and Plug-ins have been included
in the workflow followed for this study:
• Rhinoceros 3D;
• Grasshopper (parametric definition of fold pattern and target surface);
• Kangaroo (for digital simulation of fold folding);
• Karamba (for finite element analysis of folded form);
• CSI SAP2000 (for verification of FE analysis results);
• Microsoft Excel (for optimization loop).
Parameters for Structural Evaluation
The origami pattern was first defined parametrically in Grasshopper and then folded
digitally using the Kangaroo physics engine. Rigid foldability, flat-foldability and
developability properties of origami have been respected for this study. The folded
mesh was then converted to shell elements and analyzed as a concrete folded plate
structure (concrete strength = 16.7 MPa) using Karamba. The supports are all
Fig. 4 Distorted physical models of folded standard Miura tessellation
Algorithmic Design and Analysis of Architectural Origami 65
assumed to pins, restraining translation only. All loads are defined as mesh loads,
applied to mesh vertices. The self-weight is calculated as a function of the mesh
area, the thickness of the shell elements, and the unit weight of the concrete material
specified. The self-weight (SW), live load (LL), and wind load (Wx and Wy) are
applied according to the global axes, while the superimposed dead load (SDL) is
applied locally to the mesh. As an approximation of windward and leeward wind
load, half of the wind load value is applied to all of the mesh vertices. Table 2
summarizes the loads and load combinations that were used in the analysis.
Optimization
In order to compare the various origami models in terms of utilization, the
maximum utilization (calculated as Von Mises stress/material yield strength) for all
models should be approximately 1. For each model, an iterative loop evaluates the
maximum utilization of all shell elements at a given thickness. For each new
thickness, it applies an updated self-weight, and calculates the Von Mises stress in
each shell element and compares it to the material strength. If this utilization is
greater than 1 or less than 0.6, then a new thickness must be tested. If the utilization
is greater than 1, the thickness is increased by a percentage of the current input
thickness. If the utilization is less than 1, the thickness is decreased by a percentage
of the current input thickness. This iterative process continues until the following
criteria (Table 3) are met.
Table 1 Summary of analysis models
66 M. Tsiamis et al.
To avoid creating a circular reference within the Grasshopper script, the
optimization loop sends the adjusted thicknesses, which are calculated within
Grasshopper, to Excel and then reads them back into the Grasshopper script for the
next iteration. A python script, which is run in Rhino, was written to automate and
control the loop. The criteria for stopping the loop are defined in this Python script.
Figure 5 illustrates how the loop works using Grasshopper, Excel, and Python.
Initial analyses of the folded origami structures revealed that the structure was
sensitive to buckling for various combinations of loading and shell thickness
distributions. Therefore, the optimization process is based on second order analysis,
which takes into account the effect of the in-plane forces in the shell elements on their
stiffness during loading.
When the optimization routine yields a large range of different thicknesses for a
folded structure, an additional step in the routine can group the optimized
thicknesses into ranges to reduce the amount of variability. For construction, it is
preferable to have only a few different shell thicknesses in order to reduce the
number of unique elements that need to be fabricated. However, for the purpose of
comparing structural behavior, this large variety of thicknesses is acceptable.
The optimization loop was limited to 30 iterations, however, many models with
opposite edges supported did not achieve a maximum utilization of 1.0 within these 30
iterations. This was because there were often a few elements along the unsupported
edges that were overstressed, and the optimization loop continued to add material to
these elements. For future analysis, an edge beam along unsupported edges or a
diaphragm between the folds along unsupported edges will be added to the structural
model to stiffen the free edges.
Analysis Results
In the assessment of the analysis results, a model was considered to be ‘‘more
efficient’’ if more elements achieved a utilization closer to 1, if less material volume
Table 2 Load cases and combinations used in structural analysis
Loads (kN/m2) Load combinations (strength) Load combinations (deflection)
SW = Amesh*t*!concrete 1.4 (DL ? SDL) 1.0 (DL ? SDL)
SDL = 0.5 1.2 (DL ? SDL) ? 1.6LL ? 0.5Wx 1.0 (DL ? SDL) ? 1.0LL ? 1.0Wx
LL = 1.0 1.2 (DL ? SDL) ? 1.6LL ? 0.5Wy 1.0 (DL ? SDL) ? 1.0LL ? 1.0Wy
W (x and y) = 1.7 1.2 (DL ? SDL) ? 1.0LL ? 1.0Wx
1.2 (DL ? SDL) ? 1.0LL ? 1.0Wy
Table 3 Criteria for exiting optimization loop
Optimization criteria
0.6 B maximum utilization B 1
Maximum vertical displacement\ displacement limit (L/240)
Change in material volume between iterations ni and ni-1\ 10 m3
Algorithmic Design and Analysis of Architectural Origami 67
was required per unit area, and if the vertical displacement was less. A total of 37
models were analyzed for the Miura pattern to test the effect of span length,
Gaussian curvature, boundary conditions, and pattern density on the behavior of the
folded structures. Although the folded plate structures were analyzed under
combinations of vertical loading (DL, SDL, LL) and lateral wind loading, only the
analysis results for the worst-case combination of vertical loading (DL, SDL, LL)
were considered in the evaluation of structural performance. For each analysis
performed, the numerical results of element utilization, vertical displacement, and
volume per unit surface area were collected and plotted as shown in the
representative set of results for the Miura pattern in Fig. 6. Graphical results
displayed in the analysis model, including thickness distribution, utilization, vertical
displacement, force isolines, and planarity (Fig. 7) were generated and used to
qualitatively verify expected behavior under the assigned loading.
Discussion
The analysis aimed to answer the following questions: how do curvature and
boundary condition affect the ‘‘efficiency’’ of the folded structure, compared to a
shell surface of similar shape? How does pattern density and pattern direction affect
the efficiency of the folded structure?
Several conclusions can be drawn from the structural analysis of the folded Miura
tessellation with respect to the type of surface (described by its Gaussian curvature),
the degree of curvature of the target surface, the support conditions, and the density
of the pattern, as described by the number of ‘strips’ or ‘zig-zags’ in the pattern.
Shell element utilization in all cases was generally low due to sensitivity to
buckling, since buckling will occur before the element stress reaches the material
capacity.
Fig. 5 Schematic representation of optimization loop
68 M. Tsiamis et al.
Fig. 6 Analysis results plotted for Miura tessellation pulled to paraboloid target surface
Algorithmic Design and Analysis of Architectural Origami 69
Effect of Curvature on Structural Efficiency
In all three surface geometry models (paraboloid, hyperbolic paraboloid, and arch),
a lower degree of curvature in the surface shape yielded better utilization of the
panels of the folded plate structure, with more panels performing closer to their
capacity. Shallower curvature also resulted in increased deflection, as expected.
However, greater curvature resulted in the need for more material, which is not the
case for basic thin shell structures of similar shape, which utilize the increased
curvature to help resist bending forces due to loading. The need for thicker plates in
certain regions of the folded plate structure at higher degrees of surface curvature
may suggest that the plates weaken locally as they distort in order to match the
target curvature.
Effect of Support Condition on Structural Efficiency and Stability
The models were analyzed under three different support conditions: supports along
all edges, supports along zig-zag edges, and supports along strip edges. For all of
these support conditions, the paraboloid-shaped Miura tessellation consistently
required more volume per unit area to resist the applied vertical load. This suggests
that forcing the tessellation to match a surface with positive Gaussian curvature, by
bending it with positive curvature simultaneously about two orthogonal axes,
introduced distortion in the panels of the folded plate structure, making them
inherently weaker. For all support conditions, the arch-shaped and paraboloid-
shaped origami models showed similar results in terms of utilization distribution,
while the hyperbolic paraboloid model was consistently most efficient in terms of
material use (low volume/unit area). The folded plate structure showed the best
utilization distribution when forced to match the negative Gaussian curvature of the
hyperbolic paraboloid target surface. This may be attributed to the auxetic nature of
the origami tessellation.
When the strip edges were supported as opposed to the zig-zag edges, the models
for all of the surface geometries demonstrated a more even utilization distribution
(all panels were working at a similar capacity to resist the loading), but required
Fig. 7 Representative set of graphical results for qualitative verification
70 M. Tsiamis et al.
more volume per unit area and deflected more under the applied loading. This
suggests that the ‘zig-zag’ axis is the stronger axis (Fig. 8).
Deflection primarily occurred along the unsupported edges. In future analyses, a
stiffening element or additional restraints will be added to the unsupported edges in
order to obtain a better distribution of deflection over the entire structure.
Effect of Origami Pattern Geometry on Structural Efficiency
Reducing the number of zig-zag polylines in a given Miura surface pattern increases
the overall depth of the folded section and, as a result, reduces deflections.
Increasing the number of strip polylines increases the number of folds and reduces
the panel size, making the folded plate structure more efficient in terms of
utilization distribution.
Planarity of Panels
Planarity of the rigid panels in the folded origami structures is measured as the
perpendicular distance between the two diagonals of a quadrilateral surface (Fig. 9)
and provides an indication of the degree of distortion due to form-matching. A
future study on the relationship between element planarity and utilization could
provide valuable insight into construction strategies for architectural origami.
Conclusions
The ability to distort origami tessellations, such as the Miura Ori tessellation, to
match a given surface shape, paired with the inherent stiffness exhibited by folded
plate structures, gives origami tessellations great potential to be used in architectural
applications. Scaling up origami for architectural applications requires structural
Fig. 8 Analysis of different support conditions suggests strong and weak bending axis of folded platestructure
Algorithmic Design and Analysis of Architectural Origami 71
evaluation of the folded form under anticipated loading conditions and boundary
constraints, and consideration of material properties. By isolating the unit module
that makes up an origami tessellation and gaining an understanding of the geometric
relationships that define the fold pattern and of the kinematics of folding the pattern,
mathematical algorithms can be developed to generate digital representations of the
folded tessellations. This rule-based approach to generating geometric configura-
tions of folded origami embeds flexibility into the design of origami-based folded
plate structures.
In this research, a digital workflow integrated the generation of geometric models
of the folded Miura tessellation in various configurations and provided a platform
for evaluating the structural capacity of these architectural models. The results of
the structural analysis suggest that the geometry of the tessellation’s fold pattern
directly influences the structural performance of the folded plate structure. As
demonstrated by the analysis results, the functional capacity of a folded plate
structure based on a standard Miura fold pattern to resist bending is dependent on
the orientation of the pattern with respect to the support conditions of the structure.
Furthermore, the frequency of repetition of the unit module in the tessellation
directly influences the strength and stability of the structure. The efficiency of a
Miura-inspired folded plate structure is dictated by the fold pattern and therefore,
modifications to the fold pattern can be made to improve the structural performance
of the structure. It can thus be concluded that function therefore follows fold. This
research provides a framework for designing folded plate structures derived from
origami tessellations on an architectural scale and sets the stage for future work
regarding the fabrication of these structures.
References
Casale, A. and G.M. Valenti. 2013. Architettura delle superfici piegate. Le geometrie che muovono gli
origami. Rome: Edizioni Kappa.
Schenk, Mark. 2011. Folded Shell Structures. Ph.D. Thesis, University of Cambridge.
Tachi, Tomohiro. 2009. Generalization of Rigid Foldable Quadrilateral Mesh Origami. Proceedings of
the IASS Symposium 2009, 2287–2294.
Mia Tsiamis is a BIM Specialist at Leslie E. Robertson Associates (LERA). Having first joined LERA on
a temporary basis to lead a research collaboration on architectural origami with Sapienza University, Mia
Fig. 9 Measurement ofplanarity of a quadrilateralsurface
72 M. Tsiamis et al.
rejoined in November 2016 and focuses on interoperability and various research endeavors. She is a part-
time professor at the Parsons School of Design, where she teaches parametric design to architectural
design and interior design undergraduates. Some of her project collaborations and exhibitions include
‘Geometrid,’ a kinetic sculpture for the 2015 International Association for Shell and Spatial Structures
(IASS) Symposium in Amsterdam, and ‘Project EGG,’ a freeform pavilion composed of approximately
5000 modules that were 3D printed by volunteers from around the world. Mia holds a Master of Science
in Architecture from Delft University of Technology, a Professional Degree and Master of Science in
Civil Engineering from Columbia University and a Bachelor of Engineering from The Cooper Union.
Alfonso Oliva is an Associate at Leslie E. Robertson Associates (LERA) and the Director of LERA?, an
in-house Laboratory for Engineering Research and Automation. Alfonso has dedicated his career to the
pursuit of computational design, research and development in the Architecture, Engineering and
Construction (AEC) industry. Through LERA?, he collaborates with design teams on a variety of
complex projects and provides instructional seminars to other firms. As a structural engineer, he has led
the structural design of sculptural art installations, sports arenas and supertall towers. Alfonso was named
a 2016 Rising Star in Civil & Structural Engineering by Civil ? Structural Engineer and Italian Young
Talent 2016 by the Progretto Arkes Association. He has lectured at Penn State University, TU Wien,
Polytechnic University of Milan, La Sapienza University of Rome, University of Naples ‘‘Federico II’’
and University of Cassino. While earning his master’s from New York University, he was one of only 10
recipients of H2CU’s Italian National Engineering Scholarship, which fully funded his tuition and
accommodations. He holds a Master in Product Architecture from Stevens Institute of Technology, a
Master of Science in Structural Engineering from New York University-Polytechnic Institute and both a
Master and Bachelor of Civil Engineering from Universita’ degli Studi di Cassino in Italy, where he is
currently working towards a PhD in Computational Design.
Michele Calvano is an Architect, PhD in architectural representation (ssd ICAR/17) specializes in
mathematical modeling and parametric modeling. He writes articles on Reverse Modeling and on
manufacturing of the shape. He is a professor at the University Sapienza of Rome in the Master of
Science in Product Design and at the University of Camerino in the drawing classes of SAD. He also
teaches the subjects of digital drawing with mathematical software at Accademia delle Arti e Nuove
Tecnologie (http://www.accademiadellearti.it/). He is Partner at ArFacade studio (http://www.arfacade.
com/), specializing in architectural facades engineering.
Algorithmic Design and Analysis of Architectural Origami 73